Teorell instability in concentration polarization.

We investigate the development of electro-osmotic (Teorell) oscillations at a weakly charged microporous membrane without a preimposed transmembrane electrolyte concentration drop. This drop, necessary for the occurrence of oscillations, develops spontaneously as a result of concentration polarization in the solution layers adjacent to the membrane. A three-layer model comprising a membrane flanked by two diffusion layers is proposed and analyzed for galvano- and potentiostatic regimes of operation.

Effect of concentration polarization on permselectivity.

In this paper, the variation of permselectivity in the course of concentration polarization is systematically analyzed for a three-layer membrane system consisting of a nonperfectly permselective ion exchange membrane, homogeneous or heterogeneous, flanked by two diffusion layers of a binary univalent electrolyte. For a heterogeneous membrane, an ionic transport model is proposed, which is amenable to analytical treatment. In this model, assuming a constant fixed charge in the membrane and disregarding water splitting, the entire transport problem is reduced to solution of a single algebraic equation for the counterion transport number. It is concluded that for both types of membrane the concentration polarization may significantly affect the permselectivity of the system through the effects of the induced nonuniformity of the coion diffusion flux in the membrane (convexity of the coion concentration profile) and varying membrane-solution interface concentration. While the former is significant for low membrane fixed charge density, for a heterogeneous membrane, the latter might be considerably affected by the flux focusing effect at the permeable membrane segments.

Overlimiting current in a microchannel.

We revisit the classical problem of diffusion-limited ion transport to a membrane (or electrode) by considering the effects of charged sidewalls. Using simple mathematical models and numerical simulations, we identify three basic mechanisms for overlimiting current in a microchannel: (i) surface conduction carried by excess counterions, which dominates for very thin channels, (ii) convection by electro-osmotic flow on the sidewalls, which dominates for thicker channels, and (iii) transitions to electro-osmotic instability on the membrane end in very thick channels. These intriguing electrokinetic phenomena may find applications in biological separations, water desalination, and electrochemical energy storage.

Dynamics of extended space charge in concentration polarization.

This paper is concerned with ionic currents from an electrolyte solution into a charge selective solid, such as an electrode, an ion exchange membrane or an array of nanochannels in a microfluidic system. All systems of this kind have characteristic voltage-current curves with segments in which current nearly saturates at some plateau values due to concentration polarization--formation of solute concentration gradients under the passage of a dc current. A number of seemingly different phenomena occurring in that range, such as anomalous rectification in cathodic copper deposition from a copper sulfate solution, superfast vortexes near an ion-exchange granule, overlimiting conductance in electrodialysis and the recently observed nonequilibrium electro-osmotic instability, result from formation of an additional extended space charge layer next to that of a classical electrical double layer at the solid/liquid interface or, rather, from the peculiar features of the extended space charge distinguishing it from that of a common diffuse electrical double layer. In this paper we discuss the nature and origin of the extended space charge and analyze its peculiar steady state and time-dependent properties important for understanding nonequilibrium electrokinetic phenomena in ionic systems.

Extended space charge in concentration polarization.

This paper is concerned with ionic currents from an electrolyte solution into a charge-selective solid, such as, an electrode, an ion-exchange membrane or an array of nano-channels in a micro-fluidic system, and the related viscous fluid flows on the length scales varying from nanometers to millimeters. All systems of this kind have characteristic voltage-current curves with segments in which current nearly saturates at some plateau values due to concentration polarization--formation of solute concentration gradients under the passage of a DC current. A number of seemingly different phenomena occurring in that range, such as anomalous rectification in cathodic copper deposition from a copper sulfate solution, super-fast vortexes near an ion-exchange granule, overlimiting conductance in electrodialysis and the recently observed non-equilibrium electroosmotic instability, result from the formation of an additional extended space charge layer next to that of a classical electrical double layer at the solid/liquid interface. In this paper we review the peculiar features of the non-equilibrium electric double layer and extended space charge and the possibility of their direct probing by harmonic voltage/current perturbations through a linear and non-linear system's response, by the methods of electrical impedance spectroscopy and via the anomalous rectification effect. On the relevant microscopic scales the ionic transport in the direction normal to the interface is dominated by drift-diffusion; hence, the extended space charge related viscous flows remain beyond the scope of this paper.

Probing the extended space charge by harmonic disturbances.

We assess the possibility of probing the diffuse electric double layer at a permeable charge-selective interface, such as a nonblocking electrode or ion-exchange membrane, under a finite steady-state current-voltage bias by small harmonic high-frequency current-voltage disturbances. Our main conclusion is that for a finite underlimiting bias, the electric double layer at such an interface is not amenable to this kind of probe; the high-frequency response of the system is dominated by the quasielectroneutral bulk. This is similar to the previously studied zero-bias case. On the other hand, the extended space charge in such double layers may be probed in this way both by the linear and nonlinear responses, correspondingly by the method of electric impedance spectroscopy and via the previously described anomalous rectification effect. The latter appears preferable over the former as a potential experimental tool for the study of the extended space charge of a nonequilibrium electric double layer.

Bulk electroconvective instability at high Péclet numbers.

Bulk electroconvection pertains to flow induced by the action of a mean electric field upon the residual space charge in the macroscopic regions of a locally quasielectroneutral strong electrolyte. For a long time, controversy has existed in the literature as to whether quiescent electric conduction from such an electrolyte into a uniform charge-selective solid, such as a metal electrode or ion exchange membrane, is stable with respect to bulk electroconvection. While it was recently claimed that bulk electroconvective instability could not occur, this claim pertained to an aqueous, low-molecular-weight electrolyte characterized by an order-unity electroconvection Péclet number. In this paper, we show that the bulk electroconvection model transforms into the leaky dielectric model in the limit of infinitely large Péclet number. For the leaky dielectric model, conduction of the above-mentioned type is unstable, and so it is in the bulk electroconvection model for sufficiently large Péclet numbers. Such instability is sensitive to the ratio of the diffusivity of the cations to the anions. For infinite Péclet number, the case with equal ionic diffusivities is a bifurcation point separating stable and unstable regimes at the low-current limit. Further, for a cation-selective solid, when the Péclet number is finite and the anions are much more diffusive than the cations, an unreported bulk electroconvective instability is possible at low current. At higher currents and large Péclet numbers, we found that the system is unstable for all cation-to-anion diffusivity ratios, but passes from a monotonic instability to an oscillatory one as this ratio passes through unity.

Electro-convective versus electroosmotic instability in concentration polarization.

Electro-convection is reviewed as a mechanism of mixing in the diffusion layer of a strong electrolyte adjacent to a charge-selective solid, such as an ion exchange (electrodialysis) membrane or an electrode. Two types of electro-convection in strong electrolytes may be distinguished: bulk electro-convection, due to the action of the electric field upon the residual space charge of a quasi-electro-neutral bulk solution, and convection induced by electroosmotic slip, due to electric forces acting in the thin electric double layer of either quasi-equilibrium or non-equilibrium type near the solid/liquid interface. According to recent studies, the latter appears to be the likely source of mixing in the diffusion layer, leading to 'over-limiting' conductance in electrodialysis. Electro-convection near a planar uniform charge selective solid/liquid interface sets on as a result of hydrodynamic instability of one-dimensional steady state electric conduction through such an interface. We compare the results of linear stability analysis obtained for instabilities of this kind appearing in the full electro-convective and limiting non-equilibrium electroosmotic formulations. The short- and long-wave aspects of these instabilities are discussed along with the wave number selection principles.

Electroconvective instability in concentration polarization and nonequilibrium electro-osmotic slip.

This paper concerns the comparison of electroconvective instability in concentration polarization at an ion-selective membrane with previously reported nonequilibrium electro-osmotic instability. Electro-osmotic formulation represents an asymptotic limit case of the electroconvective one. An improved nonequilibrium electro-osmotic slip formula is derived. Linear stability analysis for various nonequilibrium electro-osmotic formulations is carried out, including the analytic studies of the short- and long-wave limits. The obtained results are compared with those for a full electroconvective formulation. It is observed that the shortwave singularity typical for the nonequilibrium electro-osmotic instability is removed in the full electroconvective formulation. The effect of ionic diffusivities ratio on stability is discussed.

Absence of bulk electroconvective instability in concentration polarization.

The problem of bulk electroconvective stability of quiescent electric conduction from an electrolyte solution into a charge-selective solid (ion-exchange membrane) has been revised. It is shown through a numerical solution of the linear stability problem that previously reported bulk electroconvective instability does not exist. This numerical result is supported by the short wave asymptotic analysis. Our comprehensive study confirms the result of an earlier, less detailed, report by Buchanan and Saville.

Bulk electroconvection in electrolyte.

Bulk electroconvection pertains to flow induced by the action of the mean electric field upon the residual space charge in the macroscopic regions of a locally quasielectroneutral strong electrolyte. There existed a long time controversy in the literature as to whether quiescent electric conduction from such an electrolyte into a uniform charge selective solid, such as a metal electrode or an ion exchange membrane, is stable with respect to bulk electroconvection and whether bulk electroconvection at a nonuniform solid of this type may develop into a fully fledged flow. Numerical results reported in this paper confirm previous conclusions of a linear stability analysis concerning the nonexistence of bulk electroconvective instability, while we suggest that bulk electroconvection at a nonuniform charge selective solid results in a fully fledge flow on the length scale of nonuniformity of the surface.

Wave number selection in a nonequilibrium electro-osmotic instability.

Nonequilibrium electro-osmotic slip causes instability of quiescent ionic conductance through a diffusion layer of a strong electrolyte at a charge selective solid such as ion-exchange membrane or electrode. This instability, as inferred from the outer asymptotic limit of the full singularly perturbed ionic transport problem, is of the short-wave type. This latter is a serious shortcoming of the limiting model. In this Brief Report we show that inclusion in the model of the first asymptotic corrections yields a reasonable finite wavelength selection.

Ion-exchange funneling in thin-film coating modification of heterogeneous electrodialysis membranes.

Inexpensive highly permselective heterogeneous ion-exchange membranes are prohibitively highly polarizable by a dc current for being used in electrodialysis. According to recent experiments, polarizability of these membranes may be considerably reduced by casting on their surface a thin layer of crosslinked polyelectrolyte, slightly charged with the same sign as the membrane's charge. The present paper is concerned with this effect. Concentration polarization of a permselective heterogeneous ion-exchange membrane by a dc current is determined by geometric factors, such as, the typical size of the ion-permeable "gates" at the membrane surface relative to the separation distance between them and the diffusion layer thickness. The main quantitative characteristic of polarizability of a heterogeneous membrane is its voltage/current curve with its typical saturation at the limiting current, which is lower than that for a homogeneous membrane. In the present study we modify the previously developed two-dimensional model of ionic transport in a diffusion layer at a heterogeneous ion-exchange membrane by including into consideration a homogeneous ion-exchange layer adjacent to the membrane surface. A numerical solution of the respective boundary value problem shows that, indeed, adding even a very thin and weakly charged layer of this kind increases the value of the limiting current, to that of a homogeneous membrane. What differs, for different values of coating parameters, is the form of the voltage/current curves but not the value of the limiting current, namely: the thinner is the coating and the lower the fixed charge density in it, the "slower" is the approach of the limiting current. In order to explain this feature, a simple limiting model of modified membrane is derived from the original two-layer model. In this limiting model, asymptotically valid for a thin coating, solution of the ionic transport equations in it is replaced, via a suitable averaging procedure, by a single nonlinear boundary condition for the membrane/solution interface. Rigorous analysis shows that the aforementioned property of the limiting current is an exact mathematical feature of this limiting model, when the underlying physical phenomenon is the funneling of counterions by the charged layer from the impermeable parts of the membrane towards the "entrance gates." An approximate analytical solution, developed for this model, compares well with the exact numerical one.